Q. 74.0( 9 Votes )

If the polynomial

Answer :

Formula used.


If f(x) is a polynomial with degree n


Then (x – a) is a factor of f(x) if f(a) = 0


Dividend = Divisor × Quotient + Remainder


When x + 2 is factor of polynomial


Then


x + 2 = 0


x = – 2


if x + 2 is factor of polynomial f(x) = ax3 + bx2 + x – 6


then f( – 2) is 0


f( – 2) = a( – 2)3 + b( – 2)2 + ( – 2) – 6 = 0


– 8a + 4b – 2 – 6 = 0


4b – 8a = 8


4b = 8 + 8a ……… eq 1


When x – 2 divides polynomial gives remainder 4


Then


Dividend = Divisor × Quotient + Remainder


ax3 + bx2 + x – 6 = (x – 2) × Quotient + 4


ax3 + bx2 + x – 6 – 4 = (x – 2) × Quotient


ax3 + bx2 + x – 10 = (x – 2) × Quotient


(x – 2) = 0


x = 2


if (x – 2) is factor of polynomial ax3 + bx2 + x – 10


then f(2) = 0


f(2) = a(2)3 + b(2)2 + 2 – 10


8a + 4b – 8 = 0


8a + 4b = 8 ………eq 2


Putting value of 4b from eq 1 into eq 2


8a + (8 + 8a) = 8


16a + 8 = 8


16a = 8 – 8 = 0


a = 0


Putting value of ‘a’ in eq 1


4b = 8 + 8a


4b = 8 + 8 × 0


4b = 8


b = = 2


Conclusion.


The value of a and b comes to be 0 and 2 respectively


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